Incremental RSA: string meanings and the chain rule

import Linglib.Pragmatics.RSA.Incremental · namespace RSA

Incremental rational speech act models run the pipeline word by word. In the model of Cohn-Gordon, Goodman and Potts an incomplete string gets its meaning from the complete utterances that continue it, and the probability of an utterance is the product of the speaker's word choices along it. With graded word meanings the value of an utterance is the product of the values of its words, as in the models of Waldon and Degen and of Schlotterbeck and Wang.

Main definitions

RSA.prodMeaning lex us w: the product over the words us of their values lex u w.
RSA.continuations utts pfx, RSA.stringMeaning utts m pfx: the complete utterances that continue a prefix, and the mean of the utterance meaning m over them.
RSA.chain S r ctx us: the product of a word-level speaker's choices along the words us.

Main results

RSA.prodMeaning_perm: the value of an utterance does not depend on the order of its words.

References

Declarations

The value of a word sequence is the product of the values of its words.

def prodMeaning L56–58

{U : Type u_1} {W : Type u_2} {R : Type u_3} [CommMonoid R] (lex : U → W → R) (us : List U) (w : W) :

prodMeaning lex us w = (List.map (fun (x : U) => lex x w) us).prod

Used by prodMeaning_nil, prodMeaning_cons, prodMeaning_perm, prodMeaning_nonneg, prodMeaning_le_one

Used in SchlotterbeckWang2023, WaldonDegen2021

L60

L62–64

The value of a word sequence does not depend on the order of its words.

theorem prodMeaning_perm L66–69

{U : Type u_1} {W : Type u_2} {R : Type u_3} [CommMonoid R] {lex : U → W → R} {us us' : List U} (h : us.Perm us') (w : W) :

prodMeaning lex us w = prodMeaning lex us' w

Built from

L77–79

L87–91

String meanings

The complete utterances of a closed set that continue a prefix.

def continuations L101–103

{U : Type u_1} [DecidableEq U] (utts : List (List U)) (pfx : List U) :

List (List U)

continuations utts pfx = List.filter (fun (x : List U) => decide (pfx <+: x)) utts

Used by stringMeaning, stringMeaning_pos, stringMeaning_of_continuations_eq_nil, stringMeaning_of_continuations_eq_singleton

The meaning of a prefix is the mean of the utterance meaning over the complete utterances that continue it. A prefix with no continuation has meaning zero.

stringMeaning L105–109

noncomputable def RSA.stringMeaning {U : Type u_1} {W : Type u_2} [DecidableEq U] (utts : List (List U)) (m : List U → W → ℝ) (pfx : List U) (r : W) :

ℝ

stringMeaning utts m pfx r = (List.map (fun (x : List U) => m x r) (continuations utts pfx)).sum / ↑(continuations utts pfx).length

Built from

Used by stringMeaning_nonneg, stringMeaning_pos, stringMeaning_of_continuations_eq_nil, stringMeaning_of_continuations_eq_singleton, stringMeaning_eq_of_perm_map

Used in CohnGordonEtAl2019

The chain rule

The probability that a word-level speaker S, who chooses each word given the words so far, continues the context ctx with the words us for the referent r.

chain L155–160

noncomputable def RSA.chain {U : Type u_1} {W : Type u_2} [MeasurableSpace U] [MeasurableSpace W] (S : List U → ProbabilityTheory.Kernel W U) (r : W) :

List U → List U → ℝ

chain S r x✝ [] = 1
chain S r x✝ (u :: us) = ((S x✝) r).real {u} * chain S r (x✝ ++ [u]) us

Used by chain_nil, chain_cons

Used in CohnGordonEtAl2019, WaldonDegen2021

L162–163

L165–167